Bifurcation of a kind of piecewise smooth generalized Abel equation via first and second order analyses. (English) Zbl 1465.34053
The authors investigate the bifurcation of nontrivial limit cycles in pieceswise perturbed smooth equation of the form
\[
\frac{dx}{dt}=\frac{\cos(t)}{p-1}x^p +\sum_{i=1}^{\infty} \varepsilon^i(P_i(t) x^p + Q_i(t) x^{2p-1}),
\]
where \(x\in \mathbb{R}\), \(p\in \mathbb{Z}^+\backslash \{1\}\), \(|\varepsilon|\ll1\) and \(P_i(t)\) and \(Q_i(t)\) are piecewise trigonometric polynomials defined for \(0\le t \le \pi\) and \(\pi \le t\le 2\pi\). The corresponding unperturbed system has a periodic annulus \(U=\{(t,x): x^{-p+1}+\sin(t)=\rho^{-p+1}, \rho\in(-1,1)\}\). The maximum number of limit cycles bifurcating from \(U\) is obtained in this paper by the first- and second-order Melnikov functions.
Reviewer: Yun Tian (Shanghai)
MSC:
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34A36 | Discontinuous ordinary differential equations |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34C07 | Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations |
| 37C60 | Nonautonomous smooth dynamical systems |
| 34E10 | Perturbations, asymptotics of solutions to ordinary differential equations |
References:
| [1] | Álvarez, M. J., Gasull, A. & Yu, J. [2008] “ Lower bounds for the number of limit cycles of trigonometric Abel equations,” J. Math. Anal. Appl.342, 682-693. · Zbl 1362.34068 |
| [2] | Andronov, A. A., Vitt, A. A. & Khaikin, S. E. [1966] Theory of Oscillators (Pergamon Press, Oxford-New York-Toronto) translated from the Russian by F. Immirzi; Translation edited and abridged by W. Fishwick. · Zbl 0188.56304 |
| [3] | Bernardo, M., Budd, C., Champneys, A. R. & Kowalczyk, P. [2008] Piecewise-Smooth Dynamical Systems, , Vol. 163 (Springer-Verlag, London). · Zbl 1146.37003 |
| [4] | Bostan, A. & Dumas, P. [2010] “ Wronskians and linear independence,” Amer. Math. Month.117, 722-727. · Zbl 1202.00030 |
| [5] | Cardin, P. T. & Torregrosa, J. [2016] “ Limit cycles in planar piecewise linear differential systems with nonregular separation line,” Physica D337, 67-82. · Zbl 1376.34028 |
| [6] | Cherkas, L. [1976] “ Number of limit cycles of an autonomous second-order system,” Diff. Eqs.5, 666-668. · Zbl 0365.34039 |
| [7] | Filippov, A. F. [1988] Differential Equations with Discontinuous Righthand Sides, , Vol. 18 (Kluwer Academic Publishers Group, Dordrecht) translated from the Russian. · Zbl 0664.34001 |
| [8] | Freire, E., Ponce, E., Rodrigo, F. & Torres, F. [1998] “ Bifurcation sets of continuous piecewise linear systems with two zones,” Int. J. Bifurcation and Chaos8, 2073-2097. · Zbl 0996.37065 |
| [9] | Gasull, A. & Llibre, J. [1990] “ Limit cycles for a class of Abel equations,” SIAM J. Math. Anal.21, 1235-1244. · Zbl 0732.34025 |
| [10] | Gasull, A., Li, C. & Torregrosa, J. [2012] “ A new Chebyshev family with applications to Abel equations,” J. Diff. Eqs.252, 1635-1641. · Zbl 1233.41003 |
| [11] | Han, M. & Liu, X. [2010] “ Bifurcation of limit cycles by perturbing piecewise Hamiltonian systems,” Int. J. Bifurcation and Chaos20, 1379-1390. · Zbl 1193.34082 |
| [12] | Hu, N. & Du, Z. [2013] “ Bifurcation of periodic orbits emanated from a vertex in discontinuous planar systems,” Commun. Nonlin. Sci. Numer. Simul.18, 3436-3448. · Zbl 1344.37062 |
| [13] | Huan, S. M. & Yang, X. S. [2013] “ Existence of limit cycles in general planar piecewise linear systems of saddle-saddle dynamics,” Nonlin. Anal.92, 82-95. · Zbl 1309.34042 |
| [14] | Huang, J. & Zhao, Y. [2012] “ Periodic solution for equation \(ẋ=A(t) x^m+B(t) x^n+C(t) x^l\) with \(A(t)\) and \(B(t)\) changing signs,” J. Diff. Eqs.253, 73-99. · Zbl 1277.34047 |
| [15] | Huang, J., Liang, H. & Llibre, J. [2018] “ Nonexistence and uniqueness of limit cycles for planar polynomial differential systems with homogeneous nonlinearities,” J. Diff. Eqs.265, 3888-3913. · Zbl 1405.34027 |
| [16] | Krusemeyer, M. [1988] “ Why does the wronskian work?” Amer. Math. Month.95, 46-49. · Zbl 0661.34009 |
| [17] | Li, C. & Li, W. [2010] “ Weak Hilbert’s 16th problem and relative research,” Adv. Math.39, 513-526. · Zbl 1482.34096 |
| [18] | Li, S. & Liu, C. [2015] “ A linear estimate of the number of limit cycles for some planar piecewise smooth quadratic differential system,” J. Math. Anal. Appl.428, 1354-1367. · Zbl 1327.34058 |
| [19] | Lins-Neto, A. [1980] “ On the number of solutions of the equation \(dx/dt= \sum_{j = 0}^n a_j(t) x^j, 0\leqt\leq1\), for which \(x(0)=x(1)\),” Invent. Math.59, 67-76. · Zbl 0448.34012 |
| [20] | Llibre, J. & Ponce, E. [2012] “ Three nested limit cycles in discontinuous piecewise linear differential systems with two zones,” Dyn. Contin. Discr. Impuls. Syst. Ser. B19, 325-335. · Zbl 1268.34061 |
| [21] | Llibre, J. & Mereu, A. C. [2014] “ Limit cycles for discontinuous quadratic differential systems with two zones,” J. Math. Anal. Appl.413, 763-775. · Zbl 1318.34049 |
| [22] | Loud, W. S. [1964] “ Behavior of the period of solutions of certain plane autonomous systems near centers,” Contrib. Diff. Eqs.3, 21-36. · Zbl 0139.04301 |
| [23] | Rivlin, J. T. [1990] Chebyshev Polynomials: From Approximation Theory to Algebra and Number Theory, , 2nd edition (John Wiley & Sons, NY). · Zbl 0734.41029 |
| [24] | Studden, W. J. [1966] Tchebycheff Systems: With Applications in Analysis and Statistics (Wiley). · Zbl 0153.38902 |
| [25] | Wang, Y. & Han, M. [2016] “ On the limit cycles of perturbed discontinuous planar systems with 4 switching lines,” Chaos Solit. Fract.83, 158-177. · Zbl 1355.34036 |
| [26] | Zou, C. & Yang, J. [2018] “ Piecewise linear differential system with a center-saddle type singularity,” J. Math. Anal. Appl.459, 453-463. · Zbl 1380.34048 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
